Yes. If $K$ is the set of $n\times n$ matrices with elements in a field $F$, then you can regard $K$ as a vector space over $F$. The multiplication would be multiplying every element of a matrix by the same element $\lambda\in F$. So technically you could regard the elements of $K$ as vectors.

However, matrices are rarely referred to as vectors because of the potential confusion. In particular, matrices are frequently used to represent linear transformations between vector spaces - if $A$ is a matrix in $K$ and $v$ is an ordinary $n\times 1$ column vector with elements in $F$, then $Av$ is another vector (using ordinary matrix multiplication.

The term vector can have two different meanings.
A. Most people particularly physics students & learn vectors as direction and magnitude. This meaning has an alternate representation/ viewpoint as coordinates in space (2 dimensions, 3 dimensions or higher).
In the introductory linear algebra courses I am familiar with, most examples of vectors are of this first type, so fit well as a column inside a matrix.

B. From a linear algebra & higher math perspective, the term vector has a much broader, more abstract meaning. First define a vector space, then as you mentioned a vector is an element in that space.

For example, you can have a vector space of functions, such as the vector space consisting of all polynomials of degree $3$ or less (including the zero polynomial). Addition and scalar multiplication are defined in the standard way.

Another example is the space of all continuous real valued functions defined on an interval $[a,b]$. (We can define an inner product on this space by $\langle f, g \rangle = \int_a^b f g \, dx$.)

Some clarification as to where you're coming from would be helpful to better tailor the answer to your subject. Physics, Calculus/Math of 2 or 3 dimensions, linear algebra, etc.